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Current-Driven Kink Instability in Magnetically Dominated Rotating Relativistic Jets

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Current-Driven Kink Instability in Magnetically Dominated Rotating Relativistic Jets

Yosuke Mizuno

Institute of Astronomy

National Tsing-Hua University

Collaborators

Y. Lyubarsky (Ben-Gurion Univ), P. E. Hardee (Univ. of Alabama),

K.-I. Nishikawa (NSSTC/ Univ. of Alabama in Huntsville)

Mizuno, Lyubarsky, Nishikawa, & Hardee 2012, ApJ, 757, 16

Understanding Relativistic Jets by Polarimetry and Synergetic Study from Radio to TeV, Hiroshima, Japan, March 24 – 26, 2014

Radio observation of M87 jet

- Relativistic jets: outflow of highly collimated plasma
- Microquasars, Active Galactic Nuclei, Gamma-Ray Bursts, Jet velocity ～c
- Generic systems: Compact object（White Dwarf, Neutron Star, Black Hole）+ Accretion Disk

- Key Issues of Relativistic Jets
- Acceleration & Collimation
- Propagation & Stability

- Modeling for Jet Production
- Magnetohydrodynamics (MHD)
- Relativity (SR or GR)

- Modeling of Jet Emission
- Particle Acceleration
- Radiation mechanism

- Many GRMHD simulations of jet formation (e.g., Hawley & Krolik 2006, McKinney 2006, Hardee et al. 2007) suggest that
- a jet spine (Poynting-flux jet) driven by the magnetic fields threading the ergosphere via MHD process or Blandford-Znajek process
- may be surrounded by a broad sheath wind driven by the magnetic fields anchored in the accretion disk.
- High magnetized flow accelerates G >>1, but most of energy remains in B field.

Non-rotating BH Fast-rotating BH

Spine

Sheath

Total velocity

Disk

Disk Jet/Wind

BH Jet Disk Jet/Wind

Density distribution

(McKinney 2006)

(Hardee, Mizuno & Nishikawa 2007)

- Ultra-Fast TeV flares are observed in some Blazars.
- Vary on timescale as sort as
- tv~3min << Rs/c ~ 3M9hour
- For the TeV emission to escape pair creation Γem>50is required (Begelman, Fabian & Rees 2008)
- But PKS 2155-304, Mrk 501 show “moderately” superluminal ejections (vapp ~several c)
- Emitter must be compact and extremely fast
- Model for the Fast TeV flaring
- Internal: Magnetic Reconnection inside jet (Giannios et al. 2009)
- External: Recollimation shock (Bromberg & Levinson 2009)

PKS2155-304 (Aharonian et al. 2007)

See also Mrk501, PKS1222+21

Giannios et al.(2009)

- When jets propagate outward, there are possibility to grow of two major instabilities
- Kelvin-Helmholtz (KH) instability
- Important at the shearing boundary flowing jet and external medium
- In kinetic-flux dominated jet (>103rs)

- Current-Driven (CD) instability
- Important in existence of twisted magnetic field
- Twisted magnetic field is expected jet formation simulation & MHD theory
- Kink mode (m=1) is most dangerous in such system
- In Poynting-flux dominated jet (<103rs)
Questions:

- Kelvin-Helmholtz (KH) instability
- How do jets remain sufficiently stable?
- What are the Effects & Structure of instabilities in particular jet configuration?
We try to answer the questions through 3D RMHD simulations

Modified Fast Point

Alfven Point

Slow

MS Point

Fast MS Point

Collimation Shock

Kinetic Energy Flux Dominated

with Tangled (?) Field

Modified from Graphic

courtesy David Meier

Jet Launching Region

Jet Collimation Region (10 –100 Launching Region)

Sheath

High speed spine

Poynting Flux Dominated

CD Unstable Magnetic Helicity Driven Region

Combined CD/KH

Unstable Region

KH Unstable Velocity Shear Driven Region

- Well-known instability in laboratory plasma (TOKAMAK), astrophysical plasma (Sun, jet, pulsar etc).
- In configurations with strong toroidal magnetic fields, current-driven (CD) kink mode (m=1) is unstable.
- This instability excites large-scale helical motions that can be strongly distort or even disrupt the system
- For static cylindrical force-free equilibria, well known Kruskal-Shafranov (KS) criterion
- Unstable wavelengths:
l > |Bp/Bf |2pR

- Unstable wavelengths:
- However, rotation and shear motion could significant affect the instability criterion
- Distorted magnetic field structure may trigger of magnetic reconnection. (short-time variability)

Schematic picture of CD kink instability

3D RMHD simulation of CD kink instability in PWNe (Mizuno et al. 2011)

Appl et al. (2000)

- Consider force-free field with different radial pitch profile in the rest frame of jet
- maximum growth rate: Gmax=0.133 vA/P0,
- unstable wave length: lmax=8.43P0

Wave number

Growth rate for m=-1~-4 in constant pitch case.

(P0=a in our notation:

Magnetic pitch =RBz/Bf )

Growth rate for m=-1 mode as a function of wavenumber with different pitch profile

Maximum growth rate and unstable wave number for m=-1 kink as a function of magnetic Pitch

- For relativistic force-free configuration
- Linear mode analysis provides conditions for the instability but say little about the impact instability has on the system(Istomin & Pariev (1994, 1996), Begelman(1998), Lyubarskii(1999), Tomimatsu et al.(2001), Narayan et al. (2009))
- Instability of potentially disruptive kink mode must be followed into the non-linear regime

- Helical structures have been found in Newtonian /relativistic simulations of magnetized jets formation and propagation (e.g., Nakamura & Meier 2004; Moll et al. 2008; McKinney & Blandford 2009; Mignone et al. 2010)

- Previous: we have investigated the stability and nonlinear behavior of CD kink instability in a static plasma column and axial (top-hat) jet (Mizuno et al. 2009, 2011)
- Here: we investigate the influence of jet rotation on the stability and nonlinear behavior of CD kink instability.
- We consider differentially rotating relativistic jets motivated from analytical work of Poynting-flux dominated jets (Lyubarsky 2009).
- In cylindrically equilibrium configurations (close to force-free), the poloidal and toroidal fields are comparable in the comoving frame.
- The jet structure relaxes to a locally equilibrium configuration if the jet is narrow enough (the Alfven crossing time is less than the proper propagation time).

Mizuno et al. 2006a, 2011c, & progress

- RAISHIN utilizes conservative, high-resolution shock capturing schemes (Godunov-type scheme) to solve the 3D ideal GRMHD equations (metric is static)
Ability of RAISHIN code

- Multi-dimension (1D, 2D, 3D)
- Special & General relativity (static metric)
- Different coordinates (RMHD: Cartesian, Cylindrical, Spherical and GRMHD: Boyer-Lindquist of non-rotating or rotating BH)
- Different schemes of numerical accuracy for numerical model (spatial reconstruction, approximate Riemann solver, constrained transport schemes, time advance, & inversion)
- Using constant G-law and approximate Equation of State (Synge-type)
- Parallel computing (based on OpenMP, MPI)

Mizuno et al. (2012)

- Consider: Differential rotation relativistic jet with force-free helical magnetic field
- Solving RMHD equations in 3D Cartesian coordinates
- Magnetic pitch (P=RBz/Bf): constant
- Angular velocity (W0=0,1,2,4,6)
- Density profile: decrease (r=r0 B2)
- Numerical box:-3L < x, y < 3L, 0 < z < 3L(Cartesian coordinates: 240 x 240 x 120 zones)
- Boundary: periodic in axial (z) direction
- Small velocity perturbation with m=1 andn=0.5 ~ 4 modes

Measured in comoving frame

Force-free equilibrium:

B0: magnetic amplitude

R0: characteristic radius

R0=1/4L in this work

a: pitch profile parameter

b: differential rotation parameter

a=1, b=1 in this work

Choose poloidal magnetic field:

Choose Angular velocity:

Find toroidal magnetic field:

Magnetic pitch (P= RBz/Bf) :

Jet Velocity

(Drift velocity):

solid: W0=0

dotted: W0=1

dashed: W0=2

dash-dotted: W0=4

dash-two-dotted: W0=6

Angular velocity

Toroidal field

Poloidal filed

Jet rotation velocity

Axial jet velocity

Density

Alfven velocity

W0=1

Color density contour with magnetic field lines

- Displacement of the initial force-free helical field leads to a helically twisted magnetic filament around the density isosurface with n=1 mode by CD kink instability
- From transition to non-linear stage, helical twisted structure is propagates in flow direction with continuous increase of kink amplitude.
- The propagation speed of kink ~0.1c(similar to initial maximum axial drift velocity)

solid: W0=0

dotted: W0=1

dashed: W0=2

dash-dotted: W0=4

dash-two-dotted: W0=6

Volume-averaged Kinetic energy of jet radial motion

Volume-averaged magnetic energy

Alfven crossing time

- First bump at t < 20 in Ekin is initial relaxation of system
- Initial exponential linear growth phase from t ~ 40 to t ~120 (dozen of Alfven crossing time) in all cases
- Agree with general estimate of growth rate, Gmax~ 0.1vA/R0
- Growth rate of kink instability does not depend on jet rotation velocity

- W0=2 case: very similar to W0=1 case, excited n=1 mode
- W0=4 & 6 cases:n=1 & n=2 modes start to grow near the axis region
- It is because pitch decrease with increasing W0
- In nonlinear phase, n=1 mode wavelength only excited in far from the axis where pitch is larger
- Propagation speed of kink is increase with increase of angular velocity

- In order to investigate the multiple mode interaction, perform longer simulation box cases with W0=2 & 4
- W0=2 case: n=1 & n=2 modes grow near the axis region (n=1 mode only in shorter box case)
- In nonlinear phase, growth of the CD kink instability produces a complicated radially expanding structure as a result of the coupling of multiple wavelengths
- Cylindrical jet structure is almost disrupted in long-term evolution.
- The coupling of multiple unstable wavelengths is crucial to determining whether the jet is eventually disrupted.

- The radial distribution of magnetic field in the relativistic jet is unknown.
- From the theoretical and simulation work of jet formation, we expected that the magnetic pitch is increase with increasing radius from the jet axis (poloidal field is dominated).
- Therefore, we investigate the influence of radial distribution of helical magnetic field.
Initial condition

- W0=4
- a=1, 0.75, 0.5, 0.35 <= determine how magnetic pitch is increase.

solid: s=1

dotted: a=0.75

dashed: a=0.5

dash-dotted: a=0.35

Angular velocity

Toroidal field

Poloidal field

Magnetic pitch

Axial velocity

Jet rotation velocity

Density

Sound speed

Alfven speed

solid: s=1

dotted: a=0.75

dashed: a=0.5

dash-dotted: a=0.35

Volume-averaged kinetic and magnetic energies

- When a is increasing, growth rate of CD kink instability becomes small (slow growth)
- When a becomes small, the kinetic and magnetic energies do not decrease in non-linear phase
- This indicates nonlinear stabilization and structure developed initially is maintained in the nonlinear stage (see 3D structure in next).

- a=0.75 case: growth of n=1 & n=2 axial modes around the jet axis. Nonlinear evolution is similar to a=1 case.
- a=0.5 case: growth of n=2 axial mode near the jet axis. Helical structure is slowly evolving radially.
- a=0.35 case: growth of n=2 axial mode near the axis. In nonlinear phase, helical structure does not evolve radially and maintain the structure = nonlinear evolution is saturated
- The growth of instability saturates when the magnetic pitch increases with radius = jet is stabilized.

Mizuno et al. 2014, ApJ, 784, 167

Purpose

- In previous study, we follow temporal properties (a few axial wavelengths) of CD kink instability in relativistic jets using periodic box.
- In this study, we investigate spatial properties of CD kink instability in relativistic jets using non-periodic box.
Initial Condition

- Cylindrical (top-hat) non-rotating jet established across the computational domain with a helical force-free magnetic field (mostly sub-Alfvenic speed)
- Vj=0.2c, Rj=1.0
- Radial profile: Decreasing density with constant magnetic pitch (a=1/4L)
- Jet spine precessed to break the symmetry (l~3L) to excite instability

jet

Density

+ B-field

Velocity

+B-field

- Precession perturbation from jet inlet produces the growth of CD kink instability with helical density distortion.
- Helical kink structure is advected with the flow with continuous growth of kink amplitude in non-linear phase.
- Helical density and magnetic field structure appear disrupted far from the jet inlet.

- In rotating relativistic jet case, developed helical kink structure propagates along jet axis with continuous growth of kink amplitude.
- The growth rate of CD kink instability does not depend on the jet rotation.
- The coupling of multiple unstable wavelengths is crucial to determining whether the jet is eventually disrupted in nonlinear stage.
- The strongly deformed magnetic field via CD kink instability may trigger of magnetic reconnection in the jet (need Resistive Relativistic MHD simulation).

Mizuno 2013, ApJS

- Assumption
- Consider Pestchek-type magnetic reconnection
- Initial condition
- Harris-type model（anti-parallel magnetic field）
- Anomalous resistivity for triggering magnetic reconnection （r<0.8）
- Results
- B-filed：typical X-type topology
- Density：Plasmoid
- Reconnection outflow: ~0.8c